We present a near-optimal quantum algorithm, up to logarithmic factors, for estimating the Shannon entropy in the quantum probability oracle model. Our approach combines the singular value separation algorithm with quantum amplitude amplification, followed by the application of quantum singular value transformation. On the lower bound side, we construct probability distributions encoded via Hamming weights in the oracle, establishing a tight query lower bound up to logarithmic factors. Consequently, our results show that the tight query complexity for estimating the Shannon entropy within $ε$-additive error is given by $\tildeΘ\left(\tfrac{\sqrt{n}}ε\right)$.
We present a near-optimal quantum algorithm, up to logarithmic factors, for estimating the Shannon entropy in the quantum probability oracle model. Our approach combines the singular value separation algorithm with quantum amplitude amplification, followed by the application of quantum singular value transformation. On the lower bound side, we construct probability distributions encoded via Hamming weights in the oracle, establishing a tight query lower bound up to logarithmic factors. Consequently, our results show that the tight query complexity for estimating the Shannon entropy within ε-additive error is given by Θ~(εn).
Near-term quantum devices with limited qubits motivate the study of space-bounded quantum computation in the data stream model. We show that Shannon entropy estimation exhibits an exponential separation between quantum and classical space complexity in this setting. Technically, we develop a two-stage quantum streaming algorithm based on a quantum procedure with an explicitly constructed oracle derived from the streaming input. This algorithm achieves logarithmic space complexity in the accuracy parameter over the data stream, whereas any classical streaming algorithm under the same pass complexity requires polynomial space. In sharp contrast, existing results for Shannon entropy estimation in the quantum query model achieve only a quadratic speedup. Our work establishes a natural problem with practical applications in computer networking that admits an exponential quantum space advantage, revealing a fundamental gap between quantum query complexity and streaming space complexity.
Entropy is a fundamental property of both classical and quantum systems, spanning myriad theoretical and practical applications in physics and computer science. We study the problem of obtaining estimates to within a multiplicative factor γ>1 of the Shannon entropy of probability distributions and the von Neumann entropy of mixed quantum states. Our main results are: ∙ an O(n2γ21+η)-query quantum algorithm that outputs a γ-multiplicative approximation of the Shannon entropy H(p) of a classical probability distribution p=(p1,…,pn); ∙ an O(n21+2γ21+η)-query quantum algorithm that outputs a γ-multiplicative approximation of the von Neumann entropy S(ρ) of a density matrix ρ∈Cn×n. In both cases, the input is assumed to have entropy bounded away from zero by a quantity determined by the parameter η>0, since, as we prove, no polynomial query algorithm can multiplicatively approximate the entropy of distributions with arbitrarily low entropy. In addition, we provide Ω(n3γ21) lower bounds on the query complexity of γ-multiplicative estimation of Shannon and von Neumann entropies. We work with the quantum purified query access model, which can handle both classical probability distributions and mixed quantum states, and is the most general input model considered in the literature.
We study hybrid quantum-classical algorithms in the query model. The computation consists of quantum subroutines that make at most q oracle queries; between subroutines, all qubits are measured and discarded. We prove matching upper and lower bounds for several problems in this model:
(i) (Unbiased) Phase estimation up to a precision ε requires Θ(qε21) queries;
(ii) (Unbiased) Amplitude estimation up to a precision ε requires Θ(qε2p(1−p)) queries;
(iii) Search among N items requires Θ(qN) queries;
(iv) Two level AND-OR tree (AND of m ORs, with n inputs to each OR) requires Θ(qnm) queries.
All of these bounds are optimal up to a constant factor, for all q from 1, corresponding to the classical complexity, to Q(f), the unrestricted quantum query complexity of the respective problem. We also derive a hybrid lower bound for distinguishing two distributions specified by a state-preparation unitary; this bound is tight up to a logarithmic factor.
Besides specific lower bounds, an important contribution is developing methods for proving lower bounds on hybrid quantum algorithms (which have been very ad-hoc up to now). The first two bounds follow from a common framework: we analyze probability distributions over measurement transcripts and bound a progress measure that captures their distinguishability. The AND-OR bound requires a more delicate argument, combining two progress measures that track the information available to the algorithm classically and in quantum form, respectively.
We describe a quantum algorithm to estimate the $\ensuremath{\alpha}$-Renyi entropy of an unknown density matrix $\ensuremath{\rho}\ensuremath{\in}{\mathbb{C}}^{d\ifmmode\times\else\texttimes\fi{}d}$ for $\ensuremath{\alpha}\ensuremath{\ne}1$ by combining the recent technique of quantum singular value transformations with the method of estimating normalized traces in the one clean qubit model. We consider an oracular input model where the input state is prepared via a quantum oracle that outputs a purified version of the state, assumed to be nonsingular. Our method outputs an estimate of the $\ensuremath{\alpha}$-Renyi entropy to additive precision $\ensuremath{\epsilon}$, using an expected total number $O(\frac{1}{{(x\ensuremath{\epsilon})}^{2}})$ of independent applications of a quantum circuit which coherently queries the input unitary $O(\frac{1}{\ensuremath{\delta}}log\frac{d}{\ensuremath{\epsilon}})$ times, in each case measuring a single output qubit. Here $\ensuremath{\delta}$ is a lower cutoff on the smallest eigenvalue of $\ensuremath{\rho}$ and $x=\frac{1}{d}\mathrm{Tr}({\ensuremath{\rho}}^{\ensuremath{\alpha}})$. The expected number of measurements made in this method can be compared to results in the sample complexity model that generally require $\mathrm{\ensuremath{\Theta}}(\frac{{d}^{2}}{{\ensuremath{\epsilon}}^{2}})$ samples. Furthermore, we also show that multiplicative approximations can be obtained by iteratively using additive approximations, with an overhead logarithmic in the dimension d.
Given oracle access to an unknown unitary U=eiH , the fractional query problem asks how many queries are required to implement a noninteger power Ut=eitH, 0<t<1, when the spectrum is separated from the branch cut by a gap δ. Quantum singular value transformation gives an upper bound of O(δ1logε1) queries for approximation error ε. We prove a matching lower bound for arbitrary query algorithms. Our argument reduces any N-query circuit to the approximation of eitθ by a trigonometric polynomial with degree bounded by O(N), together with Remez inequality. This allows us to establish the lower bound of Ωτ(δ1logε1). Consequently, the optimal query complexity for fractional query problem is Θτ(δ1logε1), showing that the known QSVT construction is asymptotically optimal. We also give an alternative lower bound proof based on constructing a linear functional that annihilates the approximant space, yielding a Ωτ(logε1) bound uniform to δ.
Quantum kernel methods are among the leading candidates for achieving quantum advantage in supervised learning. A key bottleneck is the cost of inference: evaluating a trained model on new data requires estimating a weighted sum ∑i=1Nαik(x,xi) of N kernel values to additive precision ε, where α is the vector of trained coefficients. The standard approach estimates each term independently via sampling, yielding a query complexity of O(N∥α∥22/ε2). In this work, we identify two independent axes for improvement: (1) How individual kernel values are estimated (sampling versus quantum amplitude estimation), and (2) how the sum is approximated (term-by-term versus via a single observable), and systematically analyze all combinations thereof. The query-optimal combination, encoding the full inference sum as the expectation value of a single observable and applying quantum amplitude estimation, achieves a query complexity of O(∥α∥1/ε), removing the dependence on N from the query count and yielding a quadratic improvement in both ∥α∥1 and ε. We prove a matching lower bound of Ω(∥α∥1/ε), establishing query-optimality of our approach up to logarithmic factors. Beyond query complexity, we also analyze how these improvements translate into gate costs and show that the query-optimal strategy is not always optimal in practice from the perspective of gate complexity. Our results provide both a query-optimal algorithm and a practically optimal choice of strategy depending on hardware capabilities, along with a complete landscape of intermediate methods to guide practitioners. All algorithms require only amplitude estimation as a subroutine and are thus natural candidates for early-fault-tolerant implementations.